Dimensional Stuff : gRANK and other Dimensionless Financial Ratios

 PEG stuff

Once upon a time I looked for a formula which incorporated the PEG Ratio in a formula that ...
>PEG Ratio?
Yes, originally called the Fool's Ratio (after the Motley Fool): PEG = (Price/Earnings)/(Earnings Growth Rate).
I think PEG stands for Price/Earnings/Growth.
For example, if P/E = 10 and Earnings Growth Rate = 20% that'd give PEG = 10/20 = 0.5
When one uses the PEG Ratio to evaluate a stock, it's usual to consider something like:

PEG < 1: the stock is a good buy
PEG > 1: the stock is NOT a good buy

>Okay, so you wanted to use the PEG Ratio to decide when to buy and when to ...
No! I just wanted to identify the dimensions of the Ratio and find a formula which contained that Ratio.
By "dimensions" I mean ... for example, consider the more familiar P/E ratio:
P/E = (Dollars per Share)/(Dollars per Share per Year) = Years.
>Huh? Years?
Yes, Years. It's sometimes interpreted as the number of Years of Earnings that you'd need to recover the stock purchase price. But let's start with something more familiar:

 Dimensional stuff

Speed is measured in kilometres per hour
Time is measured in hours
then Speed * Time = (kilometres per hour) * (hours) = kilometres

>That's like: (km / hours)*(hours) = km, right?
Yes, just cancel the "hours" ... like (2/3)*(3) = 2.
Remember, we're talking "dimensions" here. Speed * Time has the dimensions of Length. If you saw a formula which said:
Volume = Speed * Time
then you'd know something was fishy because, whereas Speed*Time is a Length, Volume has the dimensions of Length3.
On the other hand, if you saw a formula like:
Volume = Area * Speed * Time
>I'd be happy, eh?
Well, you'd probably accept it without question ... I think.
>Is there such a formula? I mean, does it actually occur anywhere?
Volume=Area*Speed*Time? Yes. Do you want to know where it occurs?
>No.

Now suppose you saw a formula like
Expected Stock Price = (Annual Earnings per Share) * (Book Value)
you'd know something was fishy because the left side has the dimensions of "dollars per share" whereas the right side ...

>The right side is (dollars per share per year)*(dollars per share) so ...
So they aren't equal. In fact, a valid formula should be dimensionally consistent. In fact, it should also work whether you measure price in dollars or yen and time in years or months and ...

>That's like Distance = Speed * Time, in Japan or the U.S or ...
Yes, in any units.

>Or e = m c2 or V = IR or F = ma or ...
Yes, yes. Anyway, although in Science our dimensions would be Length, Mass and Time, in Finance we'd take as our various "dimensions": Dollars, Shares and Years.

It's curious, don't you think? In Science one never says:
"The acceleration is 32" or "The force is 15" or "The mass is 20".
One always says:
"The acceleration is 32 ft/sec2" or "The force is 15 Newtons" or "The mass is 20 kilograms".
In Finance, one often leaves out the dimensions, eh?
>Like "The P/E Ratio is 32" and "The PEG Ratio is 0.60"
Exactly! Just think of how scary it'd be if one contemplated buying a stock with a P/E Ratio of 200 years!
>That one's already in my portfolio!
Pay attention.

>Okay, what about the dimensions of "percentage"?
Percentage? It has no dimensions. We'd measure percentage as, for example,
Percentage = 100 (change in X) / (value of X)
and the change in X has the same dimensions as X itself so ...

>So the dimensions would cancel out, eh?
Exactly.
>Uh ...it's still a bit confusing.
Okay, consider measuring the circumference of a circle and its diameter. Their ratio is π, about 3.1416, and it ...
>It's independent of what units are used, eh?
Yes. It could be feet or kilometres or light years.
>Are you talking about equations where both sides have the same dimensions or ...
Or the dimensions of financial objects? I'm talking about both. Patience.

 Financial Objects

Okay, here are some examples of financial objects and their dimensions:

• P/E Ratio = (Dollars per Share) / (Dollars per Share per Year) = Years
• PEG = (P/E) / (percentage growth per Year) = Years2
• Market Capitalization = (Shares) * (Dollars per Share) = Dollars
• Volatility (or Standard Deviation) of annual returns = percentage change in stock Price, per Year = 1/Years
• Book Value = Dollars per Share
• Earnings = Dollars per Share per Year
• Stock Return = percentage change in Price per Year = 1/Years
• Sharpe Ratio = (average Return - risk-free Return) / (Standard Deviation of Returns) = dimensionless
• DOW Index = Average of 30 stock Prices = Dollars.per Share
• S&P 500 Index = SUM of (Market Caps) = Dollars
>Hey! The Sharpe Ratio is dimensionless!
Yeah. Nice, eh?
>But don't you think that PEG, as Years2, is strange?
Yes, very strange. Whereas one can attach some significance to P/E as Years, namely the number of years that it'd take current earnings to equal the stock price, I have no idea how to interpret PEG as Years2.
>Do you know anything else measured in Years2 ?
No.
Well ... maybe, yes.
If a ball is dropped from a building, the distance it falls in T seconds is (about):
[*]     distance = (1/2)acceleration*T2
So we could write: (distance)/(acceleration) as seconds2.
>I meant financial stuff.
No.

>By the way, did you say that the DOW is the average price of 30 stocks?
Well, it started out as an average price, but if stocks split or stocks are added or deleted from the list then, in order not to introduce a sudden jump in the index, they divide by some number to keep the index the same after as before the change.
>Huh?
Okay, suppose there were just two stock in the DOW with prices \$A and \$B so DOW = (A+B)/2.
Now the first stock splits 3:1 so its investors get three of the new shares for each of their old shares. The new shares are worth \$A/3. Then DOW = (A/3+B)/2 after the split (if we didn't fiddle with this formula) BUT, to avoid a sudden change in this index, we'd insist that
(A+B)/2 = (A/3+B)/2d
where "d" is some so-called divisor. In our case, we'd have:
d = (A/3+B) / (A+B).

>For A = \$12 and B = \$8 then d = (4+8)/(12+8) = 0.60
Yes. Without that divisor our index would go (suddenly!) from (12+8)/2 = 10 to (4+8)/2 = 6.
By the way, there's a divisor for the S&P 500 and the TSE 300, too.
(A few years ago I asked Standard & Poor's for the TSE 300 divisor. It was 92,251,469,343 )
Now, can we get back to dimensional stuff?
>Be my guest.
Thanks.

 Financial Formulas
Consider a formula like:
[1]       Portfolio = P0 erT
where P0 = initial portfolio,
r = annual return (as a percentage per year) and
T = number of years.

Now, it just so happens that ex = 1 + x + x2/2 + x3/6 + ...

Note that here we're adding 1 to x to x2/2 etc. and that'd be fishy if, say, x were some length because then x2 would be an area and x3 a volume and we'd be adding a length to an area to a volume and ...
>But what if x were dimensionless?
Aha! Now you've got it! If you see a formula with ex, then the exponent x had better be dimensionless.
>But our x = r T = (percentage per year)*(years) = percentage and that's ...
Yes! It's dimensionless. Indeed, if we see a formula containing an expression like Sp, then "p" must be dimensionless as, for example, (1+r)p.

Let's consider the formula:
[2]       Portfolio = P0 (1+ r)T

Here we're adding 1 and r so r should be dimensionless (since "1" has no dimensions).
Further, the exponent T is in years, which is fishy..
We'd get this formula by starting with a portfolio of \$Po and increasing it by a fraction r each year for T years.
For the first year, we'd go from P0 to P1 = P0(1+r).
That means that r = P1/P0 - 1 = (dollars per share) / (dollars per share) - 1, a dimensionless quantity.
>But isn't r a percentage per year?
Uh ... yes, and that poses a problem, eh?

Let's change [2] into [1]:

1. Assume we calculate the gain N times per year with a fractional change of r/N each time period.
2. Then we'd have, in one year, N fractional increases, each r/N.
3. Each dollar would grow to (1+r/N) each time we do the compounding (which is N times per year).
4. That'd mean that our portfolio (originally Po) would have grown to Po (1+r/N)NT after T years
(meaning after NT time periods).
5. Letting N become infinite and noting that (1+r/N)N becomes er (as N becomes infinite), we get:
Portfolio = Po erT which is just formula [1].
That's continuous compounding and ...

>But didn't you say that, in [1], r had the dimension of 1/Time?
Yes ... yet in [2] we said it was dimensionless.
>But it's the same r!
Yes, but we're going to fix that!

Look again at P1 = P0(1+ r). We want r to have the dimensions of 1//Time yet we also want P1 / P0 to be dimensionless ... so we introduce something with the dimensions of Time, like so:
[3]       P1 = P0(1+ kr)
where "k" has the dimensions of Time (so "kr" is dimensionless).
Now, if k is a fraction of a year, say k = 1/N, then kr = r/N is the fractional change in our portfolio each time period of length 1/N years and, over T years (containing NT such time periods), our portfolio would be given by:
[4]       Portfolio = P0 (1+ r/N)NT
>What's the dimension of that N?
Remember: k = 1/N has the dimensions of Time so N has the dimensions of 1/Time so ...
>So NT is dimensionless?
Exactly! However, many people use: Portfolio = P0 (1+ r)T
What they're really using is [4], but with N = 1.
>You said "1" had no dimensions and then you say that N had the dimensions of 1/Time ... I'm completely confused!
Okay, when I said that "1" had no dimensions I lied. Sometimes it DOES have dimensions.
In fact, we can rewrite [2] as:   Portfolio = P0 (1+ r/1)1*T     where that "1" has the dimensions of 1/Time. In fact, that "1" is the number of times per year that we do the compounding (meaning the number of times per year that we apply the fractional change in our portfolio). In fact, we usually ignore it if we're considering just "1" time period per year. In fact ...
>Too many in facts! So what's your point?

My point is just this:
 [A]     Financial formulas should be dimensionally consistent

That means that both sides of an equation should have the same dimensions ... and formulas should never involve the addition of quantities with different dimensions.

 Unit Independence

Let's consider some financial entity that has the dimensions of, say, (Dollars per Share)*(Years).
>Yeah, so? Where's the dimensional inconsistency?
There is none but ...
>So, what's the problem? If it works, don't fix it.
Imagine a Japanese investor who uses some BS Ratio in order to identify stocks which are good (or bad) buys:
BS = (Book Value) / (annual Stock Return)
which has the dimensions:
BS = (Dollars per Share)/(percentage change per Year) = (Dollars per Share)*Years
He'd evaluate BS as (Yen per Share) *Years which, if there were 100 Yen to the Dollar, would be 100 times larger than a U.S. investor looking at the same stock.
>BS? What's that?
I just invented it ... to make my point.
>So what does BS stand for?
>So, now what?
So I'd sugggest that not only should financial formulas be dimensionally consistent, but that (many) financial entities are better formulated so as to be independent of the units used.
>For the PEG Ratio, what'd you suggest?
We need to consider something like:   new PEG = (P/E)/(Earnings Growth Rate) / d
where the divisor "d" has the dimensions:   d = Years2.
>Then PEG would be dimensionless, eh?
Exactly. It wouldn't depend upon which currency was used, or what time period (like Years or Months or Weeks) or whether we we're talking shares or tenths of a share or 100-share contracts or ...
>So what's "d"?
Let me cogitate, but in the meantime, we'll make a note of this:
 [B]     Many financial entities should be independent of the units used.
>Many ... not all?
No, of course not. The value of your portfolio will be in some convenient currency units: dollars or yen or francs or ...
>So what financial entities should be dimensionless?
Well ... uh, if you use a financial entity to decide when to buy or sell, maybe that entity should be dimensionless.
>Dimensionless, eh? That's important?
Who knows. I just have a hunch that they'd be useful ... like the Sharpe Ratio which is a ratio - hence a comparison - of two like quantities. After all, if you say "The P/E is low", I'd say "Low compared to what?".
Anyway, we'll see. Give me a few months to check this out, okay?
>You said the units were Dollars and Years and ...
Well, what I really mean is Currency Units and Time Units and Share Units.

 a dimensionless PEG

We're looking for something, namely "d", with the dimensions Years2, so PEG / d would be dimensionless.

We could consider:
d = 1/Volatility2 = 1/(Variance of Annual Returns) = Year2
>How about d = 1/(Annual Return)2?
But PEG already has P/E incorporated. However, maybe we could just divide PEG by another PEG to make a dimensionless PEG.
>Huh?
We'll calculate the PEG Ratio for some index and use:
d = PEG(index) = (Value of the index) / (Earnings growth rate for the index, per Year).
That way we'd be comparing the PEG for the stock against some "standard" PEG.
>But doesn't (Value of the Index) have peculiar dimensions? I mean ...
Uh ... yes, I guess so. The DOW and the S&P have different dimensions. We'd have to standardize (Value of the index) so it has the appropriate dimensions.. What we need is a method of calculating the PEG Ratio for a basket of stocks.

 the PEG of an Index
Suppose we owned ALL the stock in all the companies (say n of them) in some index.
If the market caps were M1, M2, ... Mn our portfolio would be worth M1+M2+ ... +Mn   dollars.
Suppose the total Earnings for each of the n companies were E1, E2, ... En   dollars per year.
The P/E Ratio would be (M1+M2+ ... +Mn) / (E1+E2+ ... +En)   years.
>P/E Ratio?
Yes, we'll need it to calculate a PEG. Note that P/E has the dimensions of Years ... or Time units.
Suppose that we owned a total of N shares (or Share units).
The value of our portfolio would then be
A = (M1+M2+ ... +Mn) / N   dollars per share.
>You mean Currency units per Share units?
Yes. Of course.
Further, the Earnings would be
B = (E1+E2+ ... +En) / N   dollars per share per year.

Note that, for our basket of stocks,
P/E = A / B   years.
If the Earnings for each of our n stocks grew at the annual rates: r1, r2, ... rn   per year, then our total Earnings growth (for our portfolio of n stocks) would be
C = ( r1E1+ r2E2+ ... + rnEn) / N   dollars per share per year, per year.
>"dollars per share per year, per year"? That's confusing.
Okay. You drop a ball from a tall building. Its speed increases, second by second. In fact, it increases each second by about 32 feet per second. That's the acceleration: 32 feet per second, per second, written as 32 feet per second2.

Okay, back to our PEG(index). Our earnings growth rate would then be:
D = C / B   per year
The PEG Ratio for our index would be:
PEG(index) = (A/B) / D = (A/B) / (C/B) = A/C
which has the dimensions of (dollars per share) / (dollars per share per year2) = years2.
 [C]     PEG(index) = (M1+M2+ ... +Mn) / ( r1E1+ r2E2+ ... + rnEn)   years2     where the Ms are the market caps (in dollars) and     the rs are the earnings growth rates (per year) and     the Es are the total earnings (in dollars per year)

>Remind me. Why are we doing this?
We want to introduce a normalized/standardized/dimensionless PEG Ratio for a stock, so we divide the standard PEG Ratio by the PEG Ratio for some appropriate index, like the DOW or the S&P500 or the Wiltshire5000 or ...
>Okay, okay. So do it.

 gPEG

If our stock were a large cap growth stock, we'd probably want the index to consist of large cap growth stocks. Indeed, our stock might very well be included in the index. Suppose it is. In fact, suppose it's the first stock.

The dimensionless PEG for our stock ... we'll call it the gPEG Ratio, would then be:
 [D]     gPEG(stock) = PEG(stock)/PEG(index)

Suppose there are n1 outstanding shares of our stock and the market cap is M1 dollars.
Then the price is M1/n1 dollars per share and the earnings are E1/n1 dollars per share per year.
The P/E Ratio is then (M1/n1) / (E1/n1) = M1/E1 years.
The Earnings Growth Rate is r1 per year so we get PEG(stock) = M1/(r1E1).
>Isn't that just [C], with n = 1?
Uh, yeah, I guess it is

gPEG(stock) = PEG(stock)/PEG(index) = [M1/(r1E1)] / [ (M1+M2+ ... +Mn) / ( r1E1+ r2E2+ ... + rnEn) ]

This can also be written like so:

gPEG(stock) = [M1/(M1+M2+ ... +Mn)] / [r1E1/( r1E1+ r2E2+ ... + rnEn)]

which we see is just

(market cap as a percentage of the total index market cap) /(earnings growth as a percentage of the total index growth)

>Example?
Well, suppose we're looking at the thirty stocks of the DOW and we find that a particular stock has 5% of the total market cap but 10% of the earnings growth. That'd make gPEG = 5/10 = 0.50 and ...
>And if that's the smallest gPEG, that'd be the best buy?
I didn't say that!
>Then what are you suggesting?
Me? Suggest? I just write tutorials and spreadsheets, do charts, provide food for thought ...
>Yeah, right.
Are you deep in thought?
>No! And do you realize that you don't have a single chart?
 Okay, here's comes a chart. You asked whether I could interpret the significance of the dimensions, namely Years2, for the standard PEG Ratio ... and I said I couldn't. However, for a basket of stocks, a PEG(index) for example, we can do this: For each stock, plot P/E vs Earnings Growth Rate, like Figure 1 (in blue dots). Draw the "best fit" straight line approximation to the points (in red). (That's the "regression line".) The slope of the line has the dimensions of (P/E) / (Growth Rate) = Years2. Figure 1
>So what about the dimensional significance of PEG, for a single stock?
I've already told you ... I have no idea.
>But what about bpp's suggestion, at
NFB?
Oh, yeah. That's neat. Consider the reciprocal, 1/PEG:

• 1/PEG = (Earnings Growth Rate)/(Price/Earnings) = r/(P/E) = rE/P.
• r, the Growth Rate, is measured in percentage per year.
• E, the annual earnings, is in dollars per share per year.
• So rE (in dollars per share, per year per year, ) gives the rate at which the annual dollars of earnings grow, per share.
• Dividing by the stock price P (in dollars per share) gives a neat comparison of the rate at which the earnings grow to the actual stock price ... and it's measured in 1/years2.
>Huh?
If the Earnings Per Share, E, is at E = \$0.50 and grows to \$0.60 in a year, that's a 20% growth per year so r = 0.20 and since E = \$0.50
then rE = 0.20*0.50 = 0.10 so if earnings continue to grow in this way we'd expect a growth of \$0.10 dollars per year, every year.
>Per share.
Yes, per share. But that increase per year, every year, is like that ball dropping from a tall building.
Its speed increases 32 feet per second, every second.
Now, that \$0.10 dollars per year, every year: Is it good or bad?
>For a \$2 stock, that's good: PEG = (P/E)/r = (2/0.5)/20 = 0.2. For a \$100 stock, that's bad: PEG = 10.
Exactly! So we compare rE to the stock price by dividing by P. Neat, eh?
>And that means ... what?
It means that:
• If earnings have an annual increase of rE dollars per share, every year
• then the increases would be rE, 2rE, 3rE, ... nrE after 1, 2, 3, ... n years, and
• nrE is measured in (dollars per share per year),
• so, after n years, annual earnings will have increased to E+nrE = E(1+nr) dollars per share per year,
• and that's the same as P provided E(1+nr) = P or n = [(P/E)-1]/r years = (P/E)/r - 1/r = 100PEG - 1/r.
>Why 100PEG?
Because our r is a fraction, like r = 0.20, whereas PEG uses a percentage, like 20%.
For example, if PEG = 0.8 and r = 0.25 (that's a 25% annual earnings growth) then n = 100PEG - 1/r = 80 - 4 = 76 years.
>What! 76 years! I'll be dead by then!
Yes, me too, but this is fiction. We're assuming constant earnings growth and ...
>Dimensional inconsistency! Dimensional inconsistency!
Huh?
>That equation: n = 00PEG - 1/r ? Since n is in years and PEG is in years2 and 1/r is in ...
Uh oh, yes. The left side is years, the right side is years2 + years. Uh ... let's do it differently:
• We're n years into the future now ... and the annual earnings are E(1+nr) dollars per share.
• We'd like the P/E ratio (n years in the future) to be, say , k based solely upon earnings growth ... not on stock price.
• The stock price is then still P dollars per share, the EPS is then E(1+nr) dollars per share per year.
• We want P/E(after n years) = P/[E(1+nr)] = k years ... so we can solve for n = 100PEG/k - 1/r years.
>Brilliant, so you've got dimensional consistency. Are you any smarter?
Let's do that example again:
If PEG = 0.8 and r = 0.25 (that's a 25% annual earnings growth) then n = 100PEG/k - 1/r = 80/k - 4 years.
>And if I ... uh, if you wanted a P/E of k = 1 then you'd have to wait 76 years! No wonder it was so long!
Yes, but waiting for a P/E of 8 is only 80/8 - 4 = 6 years.
>And how long until we've got a P/E of k = 40?
This is fiction, remember? However, have you found the dimensional "1" in that inconsistent equation: n = 100PEG - 1/r ?
>Yes, it's n = 100PEG/1 - 1/r. That's with k = 1.
Ain't this dimensional stuff fun?
>Not really.

 >So what about that gPEG thing? It's dimensionless, eh? It's the ratio of like entities and might also be interpreted as a slope of a line, as in Figure 2 (in green). Note that gPEG = 1 for the index itself (in red). >That's not very useful. Well ... I ... uh, we can consider gPEG as a comparison with some index. If gPEG < 1 the stock is doing better than the index. >By the way, what's the significance of the g in gPEG? It might stand for g-whiz or maybe grand or maybe golden or maybe ... >Or maybe gummy? Well ... if you insist. Figure 2

 another dimensionless PEG-type Ratio: gNUM

Now let's look for some dimensionless quantity, some number we can attach to a stock, to tell us if it's a good buy. We'd like to have a small P/E Ratio and a large Growth Rate and ...
>How about (P/E) / (Growth rate)? Just pick a stock with a small value for this ratio, eh?
Pay attention! That's just our old PEG Ratio ... and it's not dimensionless.
We'd also like a small Volatility (which some people equate to "risk" ... for some unknown reason).
>How about (P/E)*Volatility / (Growth rate)? Just pick a stock where this is small.
Let's see. It's just PEG*Volatility which has the dimensions of Years2 *(1/Years) = Years. Close, but no cigar.
>You took the words right out of my mouth!

We'll give this number a name:
 [E]     gNUM = (PEG Ratio) * Volatility2   a dimensionless appendage attached to a stock

>I assume the g stands for ...
For golden, as in the golden NUMber.
>Sure.
Our point is, even for a low-PEG stock, if the volatility is large we may be in trouble.
>Especially if Volatility = Risk, eh?
Please don't equate Volatility and Risk. Just think of Volatility as being bad. After all, if a stock has an average return of, say, R, then the annualized return (and that's MUCH more important than the average return) ... the annualized return is very nearly R - Volatility2/2 so it decreases if the Volatility increases.
>So it becomes riskier!
Not necessarily ... if R increases dramatically as well.
Anyway, don't start that risk = volatility argument again \$#@%!*?

Okay, now we calculate a gNUM for a bunch of stocks and ...
>And buy the one with the smallest gNUM!
I didn't say that! Remember, I don't give investment advice. I write tutorials, do charts ...
>Yeah, yeah. I'll tell you something. I don't like that name: gNUM.
What would you suggest?
>Remember that movie with Peter Sellers and his "Birdie Num Num"?
What would you suggest?
>I don't suggest. I nit-pick, make trouble, find your errors ...
Pay attention!

 a (dimensionless) stock rating: gRANK

Okay, here's something else of interest.
A while ago I was reading a neat article in Business Week by
Robert Barker (Oct 6, 2003) and discovered another financial entity that I'd never heard of ... and it fits right in with this dimensionless stuff.
We suppose that we're going to buy a company.

• We buy all the stock. That's the market cap \$M.
• Unfortunately, we also inherit the company's debt. That's \$D.
• Aah, but we acquire the company's cash and any marketable securities they hold. That's \$C.
• Total Net Cost: M+D-C called the Enterprise Value or EV dollars.
Note that a company with lots of debt wouldn't look too healthy, but it may have plenty of cash to cover that debt.

Okay, now we look at Earnings:

• We find the company's total annual Earnings before taxes. That's \$E.
• We add up any interest and/or revenue that the company earns from cash and securities. That's \$R.
• Total Earnings before Interest and Taxes (called EBIT) = E - R dollars per year.
Notice that, since we used the cash and securities to pay off any debt, we can't count on that source of revenue, \$R.
>Which explains why it's subracted from earnings, eh?
Exactly. However, we'll use EBITDA (which is EBIT, excluding Depreciation and Amortization).

Now we generate a modified P/E Ratio (involving a modified Market Cap and of Earnings)
We'll call it the M/E Ratio:
 [F]     M/E Ratio = EV / EBITDA   years     where EV = Enterprise Value = Mkt Cap + Debt - Cash/Securities   dollars     and EBITDA = Earnings before Interest, Taxes, Depreciation and Amortization   dollars per year

>Years? Them's the same dimensions as the ordinary, garden variety P/E ratio, eh?
Yes. Note that, were we to divide EV and EBITDA by the number of shares, we'd get:
M/E = (EV/N) / (EBITDA/N) measured in (dollars per share)/(dollars per share per year), just like the regular P/E ratio.
... but it incorporates some important financial information, don't you think?

>I assume you're going to modify the PEG Ratio ... again.
Why not?
We just use this M/E Ratio instead of the regular P/E Ratio.

However, we'll also want a dimensionless PEG-type ratio ... so we'll modify our gNUM, like so, introducing another dimensionless quantity we'll call gRANK (pronounced gee-rank).:
 [G]     gRANK = M/E * Volatility2 / (Earnings Growth Rate)   a dimensionless stock rating
The smaller the gRANK, the better.
>Why are you using Earnings Growth Rate? Why not use EBITDA Growth Rate?
I would if I could, but where would I get that data?
>I have no idea.
Well, even if company debt and cash reserves etc. aren't available, one could always go back to gNUM, eh?
>Uh ... didn't I say I don't like that name?
Pay attention.

Notice that, for a high ranking (meaning a small gRANK ) ...
>A small gRANK means a high rank?
If a stock ranks #1 that's better than a rank of #10, eh? The smaller the better.
Okay, for a high ranking (meaning a small gRANK ), we'd want small M/E and small Volatility and large Earnings Growth Rate. Notice, too, that it contains lots of info about the company: past, present and future.
>Huh?
Volatility (or Standard Deviation) is based upon past stock returns. (Volatility2 is the "Variance" of stock returns.)
M/E is a comment on the present state of the company.
Earnings Growth Rate is an estimate of future performance.

>Do you have an example?
Okay, we'll compare Microsoft and GE:
 Item MSFT GE P/E Ratio 28.6 years 20.9 years

>I'll take GE!
Pay attention! We're not finished!
 Item MSFT GE P/E Ratio 28.6 years 20.9 years EV: 227.8 B 573.6 B EBITDA 13.10 B 23.85 B M/E Ratio = EV/EBITDA 17.4 years 24.1 years

>So you're suggesting that MSFT, even with a larger P/E ratio, is the better buy?
Me? Suggest? No, I just ...
>Yeah, yeah. You just write tutorials ... and steal from a magazine!
But doesn't this stuff make you think?
>No!
Note that the reciprocal, namely EBITDA / EV (measured in 1/years), can be regarded as income from your investment.
It'd be 1/17.4 = 0.057 or 5.7% per year for MSFT and 1/24.1 = 0.041 or 4.1% per year for GE.